Note on a Generalization of Taylors Series

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PROC. N. A. S-

standard deviation varies from 8% too much to 4% too little, without the correction but from 4% too much to 8% too little if we apply it, according to the way we make up the histogram. It is doubtful if in practice w would be taken so large as 1 in such a case as this, but the dilemma would remain the same except on a reduced scale.

The illustrations have been extreme in that they possessed major discontinuities. It is probably safe to apply the Sheppard corrections in all cases of mere doubt about the contact at the ends. Whether it is worthwhile to apply the corrections at all when consideration is had for the sampling errors is another matter. For example, the relative error in a2 iS (2/n)1/' and if this is to be less than w2/12of2, the value of n must exceed 288of4/w4. On the other hand if f(x) is practically normal, the expansion for the trigonometric corrections is very rapidly convergent, as is illustrated above, and reasonably large values of w, larger than are ordinarily used in practice, give such good values of the moments that, with the Sheppard corrections, the parameters of the frequency functions may be determined well within their sampling errors, even for n large, with relatively few intervals. For example, Whittaker and Robinson, p. 189, calculate the mean and standard deviation for a set of 10,000 chest measurements given to the nearest inch. If the grouping had been in 3inch intervals we should have mean = 39.842 in place of 39.835 - 0.02 and a = 2.04 in place of 2.05 L 0.015.

NOTE ON A GENERALIZATION OF TA YLOR'S SERIES By D. V. WIDDSR* DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CHICAGO Communicated January 24, 1927

1. It is a familiar fact that if one determines the constants c0, cl . . ., cn of the polynomial Sn

(x)

=

CO

+

ClX

+ c2x2 +

. ..

+ c,,xe

in such a way that the curve y = sn(x) shall have closest contact with a curve y = f(x) at a point x = t, one obtains the first (n + 1) terms of the Taylor development of f(x),

s (x) = f(t) + f'(t)(x-t) + * *

+ fl(t) (x-t) . n!

The series Go

E [Sn(x) -S. (x)](1 -

n =O

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is then Taylor's series. Heres s,,(x) may be regarded as a linear combination of the first (n + 1) terms of the sequence

(2)

1, X, x2.... If one proceeds in a similar way with the sequence

1, sin x, cos x, sin 2x,

cos

(3)

,

2x,

one is led to a series studied by G. Teixeira' in another connection, co

E A,,[l-cos x]" + B,,[l-cos x]" sin x

f(x) 2n

An = 2

(2n)!

~

~

=

D2(D2 + 12)(D2 + 22).

.(D2 + n-12)f(O),

B =

D(D2 + 12)(D2 + 22)... .(D2 + n2)f(0), where D indicates the operation of differentiation. It seems natural then to consider the sequences (2) and (3) as special cases of a more general sequence , (4) UO(X), Ul(X), U2(X), and to investigate the nature of the series introduced by the method of approximation. employed above. The purpose of the present note is to give an outline of a general theory that can be developed along these lines, the details of which the author will publish later. 2. The functions (4) are considered to be real functions of the real variable defined in a common interval a . x _ b. Definition.-The function s,(x) = Ciui(x) s=0

is a function of approximation of order n for the point x = t if the functions ui(x) are of class CT in the neighborhood of x = t, and if sn(x) has contact of order n at least withf(x) at x = t. Denote the Wronskian of the functions vo(x), v1(x), . , v(x) by

W[vo(X), V1(x).- V,,(X)]

=

|

v/,(X)

|

and in particular for the functions of the set (4) write Wn(x) =

[o(x) U1(x)

. . .

.,u(X).

THuORSM I. If the functions f(x), uo(x), ul(x), ...., u,,(x) are of class CT in the neighborhood of x = t, and if W,,(t) $ 0, then there exists a unique function of approximation of order n for x = t.

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PNoc. N. A. S.

The series (1) now takes the form co

co

i [s.(x)-sn._(x)] = E L.f(t)g1(x, t), 1n0

(5)

no

where L4 is a linear differential operator defined by the relation L13f(x) = W[uo(x), ul(x),. ) , (X) f(x) I L. f(x) W3.. (x) and where g13(x, t) is the function of Cauchy,2

~ ~~

(t) ul(t) (t) ui' (t)

uo uot

1

gn(xp t) =W(t

()u

..

-

-U.Wt

... .U.'Wt )t... n=(). .

. U... (x) u (x) U1(X) Clearly series (5) reduces to Taylor's series for the sequence (2). THZORZM II. Let the functions f(x), uO(x), ul(x), ., u(x) be of class Cn+l, and let the Wronskians Wo(x), WI(x),., W1(x) be positive in the interval a . x . b. Then if t is a point of this interval,

UO

f(x) = : Lkf(t)gk(X, t) + R(x), k=o

where R. (x) has one of the forms

= jXgn(x, t)Ln+lf(t)dt,

R.(x)

R,. (X gM(x, g(X, t))L f(t)J gm(x, s)ds,

m .!! n,

gn

t<>JX. t<> x.

3. Define a set of functions q4k(x) by the equations

4o(X) 4~k() (X) =

=

i(x) = LWW(X)J2

Wo(x)W,

Wk(x)Wk-2(x)

[Wk.l.(x)12,2,3

k

=

2, 3,

Now introduce the Conditions A.-(a) The functions 4i(x) are of class C' in the interval a . x . b, (b) 45(x) > O, a . x. b,

lim M = 1,

(c)

mn

where

M.

=

maximum -0.(x),

m m=

minimum 0n(x) in a . x 5 b.

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THJORsM III. If the functions gbi(x) satisfy Conditions A in (a, b), and if the series co

Cngn (Xf

t),p

a

_< t _< b,

converges for a value x = xo $ t of the interval a < x _ b, then it converges absolutely in the interval x-t < xo-t J, a < x _ b and uniformly in any closed interval included therein. If the sum of the series is denoted by f(x), then

I

co

Lkf(x) = X=O E c.Lkg,(x, t),

k=

O, 1, 2., Ix-ti < Ixo-til

a<x

. b.

THZORSM IV. If the functions .i(x) satisfy Conditions A, and if

f(x) = n=o c.g,(x, u),

Ix-u

< r,

a

< x

. b,

then

f(x) = E Lxf(t)gn(x, t),

u-r < t < u + r

for all x and t satisfying the relation Ix-ti < r-It-uI, a . x . b. This theorem evidently generalizes the process of analytic continuation. THuOREM V. Let the functions ki(x) satisfy Conditions A in (a, b). Then a necessary and sufficient condition that a function f(x) defined in the interval a < x < b can be represented by a series

f(x)

=

n=o

Lf(a) g. (x, a),

a

_

x

< b

is that f(x) be the difference of two functions of class C in a . x < b,

f(x)

=

<(x) -(x)

such that L"+ (x) > 0, or +(x) =_ O; Ln+.,(X) > 0, or A.,(x) -_ O, a < x < b, n = O, 1, . This theorem is a generalization of a theorem of S. Bernstein.3 We now define Conditions B.-(a) Conditions A are satisfied in (a, b),

(b)

d

k1( >)2 0,

k= 1,2,3 ..,

n=0,1,2,..., a.x. b.

TH1ORZM VI. If conditions B are satisfied in (a, b), and if f(x) is analytic in a < x < b, then a < t < b, (6) f(x) = E Lnf(t)g1(x, t), nsO

the series being convergent in some (two-sided) neighborhood oft.

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PROC. N. A. S.

Finally, to show that the series (6) is a bonafide generalization of Taylor's series, note that for 4i(x) = 1, for which Conditions B are satisfied, (6) reduces to Taylor's series. Conditions B are also satisfied if x

0'(x)

=

e n,

and for these values one is led to a series quite different from Taylor's series. * NATIONAL RuSzE3ARCH FZLLOW. 'Bul. Sci. Math. Astr., 25, p. 200. E. Goursat, Cours d'Analyse Mathfnatique, vol. 2, p. 430. 3 Math. Ann., 75, p. 449.

2

A THEORY OF MATTER AND ELECTRICITY' By GZORGT D. BIRKHOFp DZPARTMZNT OF MATHICMATICS, HARvAR UNIVERSITY

Communicated January 22, 1927

Up to the present time no mathematical theory of matter and electricity seems to have been proposed which meets the fundamental demands of determinateness and stability. A theory which appears to satisfy these demands is presented herewith. In a second following note it is proved that this theory leads to a formula of the Balmer type for the frequencies of the small oscillations of a hydrogen atom. 1. The Perfect Fluid.-We consider first the space-time of the special theory of relativity with time coordinate xi and rectangular space coordinates x2, X3, X4, the units of length and time being so chosen that the velocity of light is 1. By an "adiabatic fluid" is meant matter whose state is determined by a functionally related pressure and density, p and p such that if ui = dxilds denotes the velocity tensor, and if the "energy tensor" is defined to be

Tii

1) puivu - pg' (g= 0,i $£j; gll = _g22 = g33 = -g44- 1), then the equations of motion are obtained by setting 0 (2) =

.=

6Xa

These equations express the relativistic form of the principles of conservation of energy (i = 1) and of linear momentum in the direction of the three axes (i = 2, 3, 4). From them it appears that a certain quantity